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Goodstein sequence
A Goodstein sequence is a certain class of integer sequences G''k''(n'').Goodstein Sequence from Wolfram MathWorld Suppose we write a nonnegative integer ''n as a sum of powers of k'' (possibly duplicated), then we write the ''k exponents themselves as similar sums of powers. For example, we can write 100 as + + = + 2}} + + 1}} + . We call this the base-''k'' hereditary representation as n''. In the above representation of 100, we can "bump" the base 2 up by one, forming a different number + 3}} + + 1}} + . We write this larger number as ''B2(100), and in general, B''[''b](n'') means finding the base-''b hereditary representation of n'' and bumping the base. Now we define the recursive sequence G''0(n'') = ''n and G''k''(n'') = ''B[k'' + 1](G''k - 1(n'')) - 1. In other words, as ''k increases, we are repeatedly bumping the base and subtracting one: :G0(100) = 100 = + 2}} + + 1}} + :G1(100) = B''2(100) - 1 = + 3}} + + 1}} + - 1 = 91507169819870 :G2(100) = ''B3(G1(100)) - 1 = + 4}} + + 1}} + 2 x + 2 x 1 - 1 = 3.486030062E156 This rapidly growing sequence is known a Goodstein sequence. Surprisingly, for all values of n'', G''k(n'') eventually peaks, declines, and returns to zero. This fact is known as '''Goodstein's theorem'. Even more surprisingly, it can be shown that Goodstein's theorem cannot be proved in Peano arithmetic. Unprovability Let G(n'') be the number of steps it takes for the Goodstein sequence starting on ''n to terminate (i.e. reach zero). Formally, G(n'') as the smallest ''k for which G''k''(n'') = 0. G(''n) is an extremely fast-growing function. Define \(R^\omega_b (n)\) to be the ordinal obtained by writing n'' in base-''b hereditary notation, then replacing every instance of b'' by \(\omega\), for instance \(100) = R^\omega_2 (2^{2^2 + 2} + 2^{2^2 + 1} + 2^2) = \omega^{\omega^\omega + \omega} + \omega^{\omega^\omega + 1} + \omega^\omega\ Then \(G(n) = f_{R^\omega_2 (n)} (3) - 3\) in the fast-growing hierarchy, which shows that \(G(n)\) is around \(f_{\varepsilon_0} (n)\). So, it grows as fast as Jonathan Bowers' tetrational arrays. Since this function grows faster than any function provably recursive in Peano arithmetic, Goodstein's theorem is not provable in Peano arithmetic. Values and bounds of \(G(n)\) Notes * \(^ba\) indicates tetration. * \(\uparrow\) indicates arrow notation. * \(Ack(n)\) indicates unary Ackermann function (Friedman's version). * \(\lbrace\rbrace\) indicates BEAF. Weak Goodstein function If we use the ordinary base representation instead of the hereditary representation, Goodstein's theorem still holds. The sequences in this system are called '''weak Goodstein sequences', producing the weak Goodstein function \(g(n)\). Like its stronger cousin, \(g(n)\) can be computed using the Hardy hierarchy, putting it on par with \(f_{\omega}(n)\) in the fast-growing hierarchy, or about \(2 \uparrow^{n-1} n\) in arrow notation. * \(g(0) = 0\) * \(g(1) = 1\) * \(g(2) = 3\) * \(g(3) = 5\) * \(g(4) = 21\) * \(g(5) = 61\) * \(g(6) = 381\) * \(g(7) = 2045\) * \(g(8) = 3 \times 2^{402653211} - 3 > 6.895 \times 10^{121210694} \) * \10^{2.0757 \times 10^{121210694}} > (g(9) > 10^{2.0756 \times 10^{121210694}} \) * \(10^{10^{10^{2.0757 \times 10^{121210694}}}} > g(10) > 10^{10^{10^{2.0756 \times 10^{121210694}}}} \) * \(10^{10^{10^{10^{10^{2.0757 \times 10^{121210694}}}}}} > g(11) > 10^{10^{10^{10^{10^{2.0756 \times 10^{121210694}}}}}} \) * \(10 \uparrow\uparrow 24 > g(12) > 10 \uparrow\uparrow 23 \) * \(10 \uparrow\uparrow 64 > g(13) > 10 \uparrow\uparrow 63 \) * \(10 \uparrow\uparrow 384 > g(14) > 10 \uparrow\uparrow 383 \) * \(10 \uparrow\uparrow 2048 > g(15) > 10 \uparrow\uparrow 2047 \) * \((10 \uparrow\uparrow)^{2} (6.896 \times 10^{121210694} > g(16) > (10 \uparrow\uparrow)^{2} (6.895 \times 10^{121210694}) \) * \((10 \uparrow\uparrow)^{3} (10^{2.0757 \times 10^{121210694}}) > g(17) > (10 \uparrow\uparrow)^{3} (10^{2.0756 \times 10^{121210694}}) \) * \((10^{10^{10^{2.0757 \times 10^{121210694}}}}) > g(18) > (10 \uparrow\uparrow)^{5} (10^{10^{10^{2.0756 \times 10^{121210694}}}}) \) * \((10^{10^{10^{10^{10^{2.0757 \times 10^{121210694}}}}}}) > g(19) > (10 \uparrow\uparrow)^{7} (10^{10^{10^{10^{10^{2.0756 \times 10^{121210694}}}}}}) \) * \(10 \uparrow\uparrow\uparrow 25 > g(20) > 10 \uparrow\uparrow\uparrow 24 \) * \((10 \uparrow\uparrow\uparrow)^{3} (g(9)) > g(32) > (10 \uparrow\uparrow\uparrow)^{3} (g(8)) \) * \((10 \uparrow\uparrow\uparrow\uparrow)^{4} (g(17)) > g(64) > (10 \uparrow\uparrow\uparrow\uparrow)^{4} (g(16)) \) In general, (10 \uparrow^{n-3})^{n-3} (g(2^{n-2})+1)\) > \(g(2^{n}) > (10 \uparrow^{n-3})^{n-3} (g(2^{n-2}))\) for \(n > 3\). Sources Category:Functions Category:Theorems